Centered regular polychoron numbers

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A001846 Centered 4-dimensional orthoplex numbers.

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Formulae

The n^th 4-dimensional N₃-cell centered regular polytope (having N₀ vertices) number is given by the formula:

$\,_{c}P_{N_{0}}^{(4)}(n)=?,\,$

where $\,$...

Recurrence equations

$\,_{c}P_{N_{0}}^{(4)}(n)=?,\,$

with initial conditions

$\,_{c}P_{N_{0}}^{(4)}(n)=?\,$

Generating function

$G_{\{\,_{c}P_{N_{0}}^{(4)}(n)\}}(x)=?,\,$

Order of basis

In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and k k-polygonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found. Joseph Louis Lagrange proved the square case (known as the four squares theorem) in 1770 and Gauss proved the triangular case in 1796. In 1813, Cauchy finally proved the horizontal generalization that every nonnegative integer can be written as a sum of k k-gon numbers (known as the polygonal number theorem,) while a vertical (higher dimensional) generalization has also been made (known as the Hilbert-Waring problem.)

A nonempty subset $\scriptstyle A\,$ of nonnegative integers is called a basis of order $\scriptstyle g\,$ if $\scriptstyle g\,$ is the minimum number with the property that every nonnegative integer can be written as a sum of $\scriptstyle g\,$ elements in $\scriptstyle A\,$. Lagrange’s sum of four squares can be restated as the set $\scriptstyle \{n^{2}|n=0,1,2,\ldots \}\,$ of nonnegative squares forms a basis of order 4.

Theorem (Cauchy) For every $\scriptstyle k\geq 3$, the set $\scriptstyle \{P(k,n)|n=0,1,2,\ldots \}\,$ of k-gon numbers forms a basis of order $\scriptstyle k\,$, i.e. every nonnegative integer can be written as a sum of $\scriptstyle k\,$ k-gon numbers.

We note that polygonal numbers are two dimensional analogues of squares. Obviously, cubes, fourth powers, fifth powers, ... are higher dimensional analogues of squares. In 1770, Waring stated without proof that every nonnegative integer can be written as a sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 1909, Hilbert proved that there is a finite number $\scriptstyle g(d)\,$ such that every nonnegative integer is a sum of $\scriptstyle g(d)\,$ $\scriptstyle d\,$^th powers, i.e. the set $\scriptstyle \{n^{d}|n=0,1,2,\ldots \}\,$ of $\scriptstyle d\,$^th powers forms a basis of order $\scriptstyle g(d)\,$. The Hilbert-Waring problem is concerned with the study of $\scriptstyle g(d)\,$ for $\scriptstyle d\geq 2\,$. This problem was one of the most important research topics in additive number theory in last 90 years, and it is still a very active area of research.

Differences

$\,_{c}P_{N_{0}}^{(4)}(n)-\,_{c}P_{N_{0}}^{(4)}(n-1)=?,\,$

Partial sums

$\sum _{n=0}^{m}\,_{c}P_{N_{0}}^{(4)}(n)=?,\,$

Partial sums of reciprocals

$\sum _{n=0}^{m}{\frac {1}{\,_{c}P_{N_{0}}^{(4)}(n)}}=?,\,$

Sum of reciprocals

$\sum _{n=0}^{\infty }{\frac {1}{\,_{c}P_{N_{0}}^{(4)}(n)}}=?,\,$

Table of formulae and values

N₀, N₁, N₂ and N₃ are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional) and cells (3-dimensional) respectively, where the cells are the actual facets. The centered regular polychorons are listed by increasing number N₀ of vertices.

Centered regular polychoron numbers formulae and values

Rank r	N0	Name (N0, N1, N2, N3) Schläfli symbol	Formulae $\,_{c}P_{N_{0}}^{(4)}(n)\,$	Generating function $G_{\{\,_{c}P_{N_{0}}^{(4)}(n)\}}(x)\,$	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	OEIS number
0	5	Pentachoron 5 cell (5, 10, 10, 5) {3, 3, 3}	$\,$	$\,$	1	6
1	8	16 cell (8, 24, 32, 16) {3, 3, 4}	$\,$	$\,$	1	9
2	16	Tesseract 8 cell (16, 32, 24, 8) {4, 3, 3}	$\,$	$\,$	1	17
3	24	24 cell (24, 96, 96, 24) {3, 4, 3}	$\,$	$\,$	1	25
4	120	600 cell (120, 720, 1200, 600) {3, 3, 5}	$\,$	$\,$	1	121
5	600	120 cell (600, 1200, 720, 120) {5, 3, 3}	$\,$	$\,$	1	601

Table of related formulae and values

N₀, N₁, N₂ and N₃ are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional) and cells (3-dimensional) respectively, where the cells are the actual facets. The centered regular polychorons are listed by increasing number N₀ of vertices.

Centered regular polychoron numbers related formulae and values

Rank r	N0	Name (N0, N1, N2, N3) Schläfli symbol	Order of basis $g_{\{\,_{c}P_{N_{0}}^{(4)}\}}=\,$	Differences $\,_{c}P_{N_{0}}^{(4)}(n)-\,$ $\,_{c}P_{N_{0}}^{(4)}(n-1)=\,$	Partial sums $\sum _{n=0}^{m}{\,_{c}P_{N_{0}}^{(4)}(n)}=$	Partial sums of reciprocals $\sum _{n=0}^{m}{1 \over {\,_{c}P_{N_{0}}^{(4)}(n)}}=$	Sum of Reciprocals $\sum _{n=0}^{\infty }{1 \over {\,_{c}P_{N_{0}}^{(4)}}}=$
0	5	Pentachoron 5 cell (5, 10, 10, 5) {3, 3, 3}	$\,$	$\,$	$\,$	$\,$	$\,$
1	8	16 cell (8, 24, 32, 16) {3, 3, 4}	$\,$	$\,$	$\,$	$\,$	$\,$
2	16	Tesseract 8 cell (16, 32, 24, 8) {4, 3, 3}	$\,$	$\,$	$\,$	$\,$	$\,$
3	24	24 cell (24, 96, 96, 24) {3, 4, 3}	$\,$	$\,$	$\,$	$\,$	$\,$
4	120	600 cell (120, 720, 1200, 600) {3, 3, 5}	$\,$	$\,$	$\,$	$\,$	$\,$
5	600	120 cell (600, 1200, 720, 120) {5, 3, 3}	$\,$	$\,$	$\,$	$\,$	$\,$

Table of sequences

Centered regular polychoron numbers sequences

N0	$\,_{c}P_{N_{0}}^{(4)}(n),\ n\geq 0\,$ sequences
5	{1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
8	{1, 9, ...}
16	{1, 17, ...}
24	{1, 25, ...}
120	{1, 121, ...}
600	{1, 601, ...}

See also

• Regular polychoron numbers

External links

• S. Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
• S. Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
• Herbert S. Wilf, generatingfunctionology, 2^nd ed., 1994.